![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cnvss | GIF version |
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
cnvss | ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3055 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (〈𝑦, 𝑥〉 ∈ 𝐴 → 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
2 | df-br 3894 | . . . 4 ⊢ (𝑦𝐴𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) | |
3 | df-br 3894 | . . . 4 ⊢ (𝑦𝐵𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3imtr4g 204 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥)) |
5 | 4 | ssopab2dv 4158 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
6 | df-cnv 4505 | . 2 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
7 | df-cnv 4505 | . 2 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
8 | 5, 6, 7 | 3sstr4g 3104 | 1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1461 ⊆ wss 3035 〈cop 3494 class class class wbr 3893 {copab 3946 ◡ccnv 4496 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-in 3041 df-ss 3048 df-br 3894 df-opab 3948 df-cnv 4505 |
This theorem is referenced by: cnveq 4671 rnss 4727 relcnvtr 5014 funss 5098 funcnvuni 5148 funres11 5151 funcnvres 5152 foimacnv 5339 tposss 6095 structcnvcnv 11812 |
Copyright terms: Public domain | W3C validator |